Kirsten and Shelicia are making a quilted wall hanging. They are cutting out \begin{align*}4^{{\prime}{\prime}}\end{align*} square pieces, and will sew them into a quilt that is 10 rows by 10 rows. Once the quilt pieces are sewn together, the squares will be a bit smaller because of the seams.

Estimate the size of the finished quilt.

In this concept, you will learn how to estimate squares and square roots.

### Estimating Square Roots

You may already know that a **perfect square** is a number whose square root is a whole number. You also know that every number has a square root, and most numbers are not perfect squares. If you need to find the square root of a number that is not a perfect square, you can determine which two whole numbers the root falls between.

Let’s look at an example.

Find \begin{align*}\sqrt{30}\end{align*}

First, you know that 30 is not a perfect square because there is no whole number you can multiply by itself to equal 30.

Next, you know the following perfect squares.

\begin{align*}\begin{array}{rcl} 5 \times 5 &=& 25 \\ 6 \times 6 &=& 36 \end{array}\end{align*}

The answer is \begin{align*}\sqrt{30}\end{align*} is between 5 and 6.

Another way of approximating the square root is to use a calculator. All calculators have a radical sign on them. Some of them may work differently, but the procedure below should be similar.

Find \begin{align*}\sqrt{30}\end{align*} using a calculator.

First, enter the number you would like to find the square root of, in this case, 30.

Next, enter the radical sign.

Then, press the equal sign.

The answer is 5.477225575

When you’re doing an estimate and writing it in an equation, it’s best to use the \begin{align*}\approx\end{align*} symbol, which means “approximately equal to.”

\begin{align*}\sqrt{30} \approx 5.477225575\end{align*}

You can also round your answer to a decimal.

\begin{align*}\sqrt{30} \approx 5.5\end{align*}

As you can see, 5.5 is between 5 and 6.

You can check the answer by squaring 5.5. Remember, this is an estimate that has also been rounded.

\begin{align*}5.5 \times 5.5=30.25\end{align*}

And \begin{align*}30.25 \approx 30\end{align*}

Your calculator has a \begin{align*}x^2\end{align*} key to help you square decimal numbers.

Estimate the value:

\begin{align*}8.2^2 \end{align*}

First, you know that 8 is the root of the perfect square 64.

Next, enter the number 8.2

Then hit the \begin{align*}x^2\end{align*} key.

The answer is 67.24.

\begin{align*}8.2^2 = 67.24\end{align*}

Since the answer on the calculator contains only two decimal places, this is an exact answer and does not require the \begin{align*}\approx\end{align*} sign.

The answer is 67.24.

### Examples

#### Example 1

Earlier, you were given a problem about Kirsten and Shelicia, who were trying to hang a quilt on the wall.

The girls are sewing together \begin{align*}4^{{\prime}{\prime}}\end{align*} squares of fabric into a \begin{align*}10 \ \text{piece} \times 10 \ \text{piece}\end{align*} quilt but some of the length will be lost due to the seams.

First, use the formula:

\begin{align*}A=s^2\end{align*}

Next, fill in what you know, and perform the calculations according to the order of operations.

\begin{align*}\begin{array}{rcl} A &=& (10 \times 4)^2 \\ A &=& 40^2 \\ A &=& 160 \ sq \ in \end{array}\end{align*}

Then, you know that 160 is a perfect square, and the girls’ quilt will be somewhat smaller once the squares are sewn together.

The answer is \begin{align*}A \approx 160 \ sq \ in\end{align*}.

#### Example 2

Estimate \begin{align*}\sqrt{11}\end{align*} to two decimals

First, recognize that 11 is between the perfect squares of 9 and 16, so you know that the answer is somewhere between their roots of 3 and 4, respectively.

Next, use your calculator.

\begin{align*}\sqrt{11} \approx 3.31662479\end{align*}

Then round to two decimal places.

The answer is \begin{align*}\sqrt{11} \approx 3.32\end{align*}.

The estimation of a number between 3 and 4 is correct.

Let’s look at another example.

Estimate the value of \begin{align*}4.9^2\end{align*}.

First, you know that 4.9 is very close to 5 and the perfect square of 25.

Next, use your calculator.

\begin{align*}4.9^2 = 24.01\end{align*}

The answer is 24.01.

#### Example 3

What is \begin{align*}\sqrt{99}\end{align*} to two decimal places?

First, you know that 99 is very close to 100, the perfect square of \begin{align*}10 \times 10\end{align*}.

Next, use your calculator.

\begin{align*}\sqrt{99} \approx 9.949874371\end{align*}

Then round to two decimal places.

The answer is \begin{align*}\sqrt{99} \approx 9.99\end{align*}

That is very close to 10.

#### Example 4

Brad wants to paint a checkerboard on a square table that covers 625 square cm. There are eight squares per side on the playing board and Brad is making each square \begin{align*}3 \ cm \times 3 \ cm\end{align*}. Brad has estimated that the checkerboard will fit. Is he correct?

First, visualize a checkerboard.

Next, you know that \begin{align*}A=s^2\end{align*}

Substitute what you know into the equation.

\begin{align*}A= (3 \times 8)^2 \end{align*}

Remembering the order of operations, multiply.

\begin{align*}A= (24)^2\end{align*}

The sides of the checkerboard will measure 24 cm each.

Then, square.

\begin{align*}A=576 \ sq \ cm\end{align*}

The answer is that the checkerboard will fit. Brad was correct.

#### Example 5

Approximate this square root to the nearest hundredth.

\begin{align*}\sqrt{65}\end{align*}

First, recognize that 65 is not a perfect square.

Next, use your calculator.

\begin{align*}\sqrt{65} \approx 8.062257748\end{align*}

Then, round to hundredths.

\begin{align*}\sqrt{65} \approx 8.06 \end{align*}

The answer is 8.06.

### Review

Find the approximate square root and round to the nearest tenth.

- \begin{align*}\sqrt{8}\end{align*}
- \begin{align*}\sqrt{11}\end{align*}
- \begin{align*}\sqrt{24}\end{align*}
- \begin{align*}\sqrt{31}\end{align*}
- \begin{align*}\sqrt{37}\end{align*}
- \begin{align*}\sqrt{43}\end{align*}
- \begin{align*}\sqrt{59}\end{align*}
- \begin{align*}\sqrt{67}\end{align*}
- \begin{align*}\sqrt{73}\end{align*}
- \begin{align*}\sqrt{80}\end{align*}
- \begin{align*}\sqrt{95}\end{align*}
- \begin{align*}\sqrt{99}\end{align*}
- \begin{align*}\sqrt{101}\end{align*}
- \begin{align*}\sqrt{150}\end{align*}
- \begin{align*}\sqrt{136}\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 9.5.